ELEC 3004 Quiz 1 Review PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Fri, 30 Mar 2012 09:52:21 UTC Contents Articles Signals 1 Signal (electronics) 1 Even and odd functions 3 Linear system 6 Time-invariant system 8 Causal system 10 Sampling 12 Sampling (signal processing) 12 Nyquist–Shannon sampling theorem 16 Aliasing 29 Quantization (signal processing) 34 Convolution 42 Convolution 42 Appendix: Systems & Laplace 53 Ordinary differential equation 53 Laplace transform applied to differential equations 62 Fourier transform 63 Harmonic oscillator 83 References Article Sources and Contributors 94 Image Sources, Licenses and Contributors 96 Article Licenses License 97 1 Signals Signal (electronics) A signal is any stream of quantities in time or spatial sequence. Signals categorizes to the fields of communications, signal processing, and to electrical engineering more generally. Signals may contain and transport information in coded form like modulation. In that case the actual quantities are finally not used, but are decoded in a detector or demodulator. In the physical world, any quantity measurable through time or over space can be taken as a signal. Within a complex society, any set of human information or machine data can also be taken as a signal. Such information or machine data (for example, the dots on a screen, the ink making up text on a paper page, or the words now flowing into the reader's mind) must all be part of systems existing in the physical world – either living or non-living. Despite the complexity of such systems, their outputs and inputs can often be represented as simple quantities measurable through time or across space. In the latter half of the 20th century, electrical engineering itself separated into several disciplines, specializing in the design and analysis of physical signals and systems, on the one hand, and in the functional behavior and conceptual structure of the complex human and machine systems, on the other. These engineering disciplines have led the way in the design, study, and implementation of systems that take advantage of signals as simple measurable quantities in order to facilitate the transmission, storage, and manipulation of information. Some definitions Definitions specific to subfields are common. For example, in information theory, a signal is a codified message, that is, the sequence of states in a communication channel that encodes a message. In the context of signal processing, arbitrary binary data streams are not considered as signals, but only analog and digital signals that are representations of analog physical quantities. In a communication system, a transmitter encodes a message into a signal, which is carried to a receiver by the communications channel. For example, the words "Mary had a little lamb" might be the message spoken into a telephone. The telephone transmitter converts the sounds into an electrical voltage signal. The signal is transmitted to the receiving telephone by wires; and at the receiver it is reconverted into sounds. In telephone networks, signalling, for example common-channel signaling, refers to phone number and other digital control information rather than the actual voice signal. Signals can be categorized in various ways. The most common distinction is between discrete and continuous spaces that the functions are defined over, for example discrete and continuous time domains. Discrete-time signals are often referred to as time series in other fields. Continuous-time signals are often referred to as continuous signals even when the signal functions are not continuous; an example is a square-wave signal. A second important distinction is between discrete-valued and continuous-valued. Digital signals are sometimes defined as discrete-valued sequencies of quantified values, that may or may not be derived from an underlying continuous-valued physical process. In other contexts, digital signals are defined as the continuous-time waveform signals in a digital system, representing a bit-stream. In the first case, a signal that is generated by means of a digital modulation method is considered as converted to an analog signal, while it is considered as a digital signal in the second case.signal transfers information Signal (electronics) 2 Discrete-time and continuous-time signals If for a signal, the quantities are defined only on a discrete set of times, we call it a discrete-time signal. In other words, a discrete-time real (or complex) signal can be seen as a function from (a subset of) the set of integers to the set of real (or complex) numbers. A continuous-time real (or complex) signal is any real-valued (or complex-valued) function which is defined for all time t in an interval, most commonly an infinite interval. Analog and digital signals Less formally than the theoretical distinctions mentioned above, two main types of signals encountered in practice are analog and digital. In short, the difference between them is that digital signals are discrete and quantized, as defined below, while analog signals possess neither property. Discretization One of the fundamental distinctions between different types of signals is between continuous and discrete time. In the mathematical abstraction, the domain of a continuous-time (CT) signal is the set of real numbers (or some interval thereof), whereas the domain of a discrete-time (DT) signal is the set of integers (or some interval). What these integers represent depends on the nature of the signal. DT signals often arise via sampling of CT signaexample, consists of a continually fluctuating voltage on a line that can be digitized by an ADC circuit, wherein the circuit will read the voltage level on the line, say, every 50 microseconds. The resulting stream of numbers is stored as digital data on a discrete-time signal. Computers and other digital devices are restricted to discrete time. Quantization If a signal is to be represented as a sequence of numbers, it is impossible to maintain arbitrarily high precision - each number in the sequence must have a finite number of digits. As a result, the values of such a signal are restricted to belong to a finite set; in other words, it is quantized. Examples of signals • Motion. The motion of a particle through some space can be considered to be a signal, or can be represented by a signal. The domain of a motion signal is one-dimensional (time), and the range is generally three-dimensional. Position is thus a 3-vector signal; position and orientation is a 6-vector signal. • Sound. Since a sound is a vibration of a medium (such as air), a sound signal associates a pressure value to every value of time and three space coordinates. A microphone converts sound pressure at some place to just a function of time, generating a voltage signal as an analog of the sound signal. Sound signals can be sampled to on a discrete set of time points; for example, compact discs (CDs) contain discrete signals representing sound, recorded at 44,100 samples per second; each sample contains data for a left and right channel, which may be considered to be a 2-vector signal (since CDs are recorded in stereo). • Images. A picture or image consists of a brightness or color signal, a function of a two-dimensional location. A 2D image can have a continuous spatial domain, as in a traditional photograph or painting; or the image can be discretized in space, as in a raster scanned digital image. Color images are typically represented as a combination of images in three primary colors, so that the signal is vector-valued with dimension three. • Videos. A video signal is a sequence of images. A point in a video is identified by its two-dimensional position and by the time at which it occurs, so a video signal has a three-dimensional domain. Analog video has one continuous domain dimension (across a scan line) and two discrete dimensions (frame and line). Signal (electronics) 3 • Biological membrane potentials. The value of the signal is a straightforward electric potential ("voltage"). The domain is more difficult to establish. Some cells or organelles have the same membrane potential throughout; neurons generally have different potentials at different points. These signals have very low energies, but are enough to make nervous systems work; they can be measured in aggregate by the techniques of electrophysiology. Entropy Another important property of a signal (actually, of a statistically defined class of signals) is its entropy or information content. References • Hsu, P. H. Schaum's Theory and Problems: Signals and Systems, McGraw-Hill 1995, ISBN 0-07-030641-9 • Lathi, B.P., Signal Processing & Linear Systems, Berkeley-Cambridge Press, 1998, ISBN 00941413357 • Shannon, C. E., 2005 [1948], "A Mathematical Theory of Communication," (corrected reprint [1]), accessed Dec. 15, 2005. Orig. 1948, Bell System Technical Journal, vol. 27, pp. 379–423, 623-656. References [1] http:/ / cm. bell-labs. com/ cm/ ms/ what/ shannonday/ paper. html Even and odd functions In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function f(x) = xn is an even function if n is an even integer, and it is an odd function if n is an odd integer. Even functions Let f(x) be a real-valued function of a real variable. Then f is even if the following equation holds for all x in the domain of f: Geometrically speaking, the graph face of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis.